Modes of convergence of Markov chain transition probabilities

Abstract

Suppose {Pn(x, A)} denotes the transition law of a general state space Markov chain {Xn}. We find conditions under which weak convergence of {Xn} to a random variable X with law L (essentially defined by ∝ Pn(x, dy) g(y) → ∝ L(dy) g(y) for bounded continuous g) implies that {Xn} tends to X in total variation (in the sense that ∥ Pn(x, .) - L ∥ → 0), which then shows that L is an invariant measure for {Xn}. The conditions we find involve some irreducibility assumptions on {Xn} and some continuity conditions on the one-step transition law {P(x, A)}. © 1977.